1. **State the problem:** Simplify or factor the quadratic expression $6x^2 - 13x - 5$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate product and sum:** Here, $a=6$, $b=-13$, $c=-5$. Calculate $a \times c = 6 \times (-5) = -30$. 4. Find two numbers that multiply to $-30$ and add to $-13$. These numbers are $-15$ and $2$ because $-15 \times 2 = -30$ and $-15 + 2 = -13$. 5. Rewrite the middle term using these numbers: $$6x^2 - 15x + 2x - 5$$ 6. Group terms: $$ (6x^2 - 15x) + (2x - 5) $$ 7. Factor each group: $$ 3x(2x - 5) + 1(2x - 5) $$ 8. Factor out the common binomial: $$ (3x + 1)(2x - 5) $$ 9. **Final answer:** The factorization of $6x^2 - 13x - 5$ is $$ (3x + 1)(2x - 5) $$